Frequently Asked Questions In Quantitative Finance

Chapter 10: Brainteasers 369

the taxi was green the witness will mistakenly say that
the car is blue 20% of the time, i.e. 17 times. In the 15
blue cases the witness will correctly say blue 80% of
the time, i.e. 12 times. So although there were only 15
accidents involving a blue taxi there were 29 reports of
a blue taxi being to blame, and most of those (17 out of
29) were in error. These are the so-called false positives
one gets in medical tests.

Now, given that we were told it was a blue taxi, what is
the probability that it was a blue taxi? That is just 12/29
or 41.4%.

Annual returns

Every day a trader either makes 50% with probability 0.6
or loses 50% with probability 0.4. What is the probability
the trader will be ahead at the end of a year, 260 trading
days? Over what number of days does the trader have
the maximum probability of making money?

(Thanks to Aaron.)

Solution
This is nice one because it is extremely counterintuitive.
At first glance it looks like you are going to make money
in the long run, but this is not the case.

Letnbe the number of days on which you make 50%.
After 260 days your initial wealth will be multiplied by

1. 5 n 0. 5260 −n.

So the question can be recast in terms of findingnfor
which this expression is equal to 1:

n=

260 ln 0. 5

ln 05.−ln 1. 5

=

260 ln 2

ln 3

= 164. 04.